Everth Cabrera, Chris Denorfia and Kyle Blanks all connected off Kershaw (2-2), who entered with a 1.16 ERA in three starts. But the 2011 NL Cy Young Award winner left this one trailing 5-1 after throwing 109 pitches in 5 1/3 innings.
After arriving with a 2-10 record, San Diego earned its first sweep at Dodger Stadium of three or more games since July 2006.
"It was big, no doubt," Padres manager Bud Black said. "It started with the starting pitching. This series was a good sign. Offensively, some guys are getting closer to their marks. I like the fact that we caught the ball. ... It was a well-played series by us, with some guys starting to swing the bats."
It was the first series between the NL West rivals since their wild brawl last week that left Los Angeles pitcher Zack Greinke with a broken collarbone. Padres slugger Carlos Quentin sat out the series while serving an eight-game suspension for charging and tackling Greinke after getting hit by a pitch.
Cabrera had three of San Diego's 13 hits and scored twice. Blanks added two hits and two RBIs as the Padres sent Los Angeles to its fourth straight defeat.
"We have to move forward," said manager Don Mattingly, who addressed his club after the game. "The way I do it is through conversation and talks and things like that. We're 15 games in, we're 7-8, and that's not good enough. But also, it's not like we're throwing up the white flag yet. We talk about what's acceptable and what's not acceptable and how do we turn it around."
Headley was back at third base after missing the first 14 games of the season with a broken left thumb. He went 1 for 4 with a late RBI single and scored a run from his No. 3 spot in the batting order.
The 28-year-old slugger had a breakout season last year, hitting 31 homers with an NL-best 115 RBIs while winning his first Gold Glove and Silver Slugger awards. He finished fifth in MVP voting.
"When a bunch of guys are hitting I think the other hitters relax," Black said. "When you're not hitting, guys have a tendency to say, `I've got to get on base, I've got to do a little bit more to help our club.' When the group is hitting, there's a little bit of pressure off everybody."
Injured while sliding into second base trying to break up a double play in a March 17 spring training game, Headley was activated from the disabled list before Wednesday's game.
To clear a roster slot, the Padres placed center fielder Cameron Maybin on the 15-day DL, retroactive to April 16, with a sore right wrist.
Kershaw yielded seven hits, walked four and struck out five. He was charged with five runs, three earned.
The left-hander racked up 25 strikeouts with only four walks and 12 hits allowed in 23 1/3 innings over his first three outings.
"You've got to be ready to hit," Black said. "Because if you get behind, you're in trouble. He has too many weapons to put you away."
Early on, Kershaw appeared to be on top of his game as he retired his first eight batters. But after opposing pitcher Tyson Ross hit a line drive off the center-field fence, things suddenly unraveled for the Dodgers.
The big inning for the Padres was the fourth, when Cabrera led off with a homer and they scored three times. They had three hits in the inning, but Kershaw also walked three (one intentional) and third baseman Nick Punto dropped a foul popup for an error that helped fuel the rally.
The Dodgers offense, with a struggling Matt Kemp beginning the game on the bench to clear his head, couldn't get much going against Ross until the fifth, when they loaded the bases with two outs. Black replaced Ross with Brad Brach (1-0), who retired Skip Schumaker on a comebacker.
The Dodgers scored their only run off Ross on a sacrifice fly by Adrian Gonzalez. Kemp entered in the seventh as pinch hitter, batting for Schumaker with the bases loaded, but struck out. Kemp came up again in the ninth with the bases loaded and drove in a run with a sacrifice fly.
"We're not going to sit and feel sorry for ourselves in any way, shape or form," Mattingly said.
Kemp was rested by Mattingly, who said he could see Kemp's "frustration" at the plate. Kemp began the night hitting .185 with no homers, four RBIs and 17 strikeouts. Schumaker started in center field in Kemp's place. ... The Dodgers placed pitcher Chris Capuano on the 15-day disabled list with a strained left calf. The injury supposedly occurred Tuesday against the Padres when he ran to cover first base. But one published report indicated that Capuano first injured his calf when running in from the bullpen during the brawl with the Padres last week in San Diego. Capuano had taken Greinke's rotation spot. Ted Lilly, currently on a minor league rehab assignment while recovering from shoulder surgery, is expected to take Capuano's place in the rotation, but the Dodgers will not need a fifth starter until April 24 because of off days Thursday and Monday next week. ... Replacing Capuano on the roster was catcher Tim Federowicz, who was hitting .533 (8 for 15) at Triple-A Albuquerque. ... Dodgers utility player Jerry Hairston Jr., suspended for one game for his role in last Thursday's fight with the Padres, decided to drop his appeal and served the suspension Wednesday.